We use the strong hypothesis that \(H_1\)  and \(H_2\)  are continuously differentiable. Indeed, in this case one has, thanks to the Envelope theorem (see [1]}),
\(\left\lbrace \begin{array}{ll}\partial _p H_1(x, -D\tilde{\phi }(t,x)) = b(x, \tilde{\alpha }^1(t,x)) \\\partial _M H_2(x, -D^2\tilde{\phi }(t,x)) = \sigma ^T\sigma (x, \tilde{\alpha }^2(t,x)).\end{array}\right.\) 
